How do I evenly wrap a mesh around a sphere
You don't tell a lot about why, for example, you begin with a flat shape, is it necessary? Anyway, I guess there are several ways to get the final shape. Here I tried Cast and Array modifiers.

If you need to start with a flat shape, you could do it with the CAST modifier:


Create your star mesh.





In Object mode, create a circle a bit above the star, made of some concentric edges loops, keep only the edges. Choose top orthographic view, select the circle, then the star, go in Edit mode and in the bottom menu bar choose Mesh > Knife Project, it's going to print the circle edges on the star geometry.





Now your star has enough geometry for a nice deformation (here it's not optimal but you can rearrange it).





Create an empty at the same point as the star origin. Give your star a Cast modifier with the empty as Control Object.





Rise the empty a bit, play with the Cast Factor, the star is going to deform and shape as a sphere!!!!!!





Keep a copy of your star + empty objects, apply the Cast modifier, to make your shape even more spherical you can shrinkwrap it.




You could do it with the ARRAY modifier:


Create a UV Sphere. Keep only a segment.





Create an empty at the sphere's origin point. Give your object an Array modifier / Object Offset mode, with the empty as Object.





Choose the right array count, rotate the empty on the Z axis so that it recreate a complete sphere. Go in Edit mode to edit your segment.





Edit your mesh until you get the result you want.